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If A is a $k\times k$ matrix,B is a $k\times l$ matrix and C is a $l\times l$ matrix prove that:

$\det{\begin{bmatrix}A&B\\O&C\end{bmatrix}}=\det(A)\det(C)$

O is the matrix that all it's elements are equal to zero.

I know some rules for calculating determinants but I don't know how to begin in this question.

K.N
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  • See e.g. https://math.stackexchange.com/questions/1905652/proofs-of-determinants-of-block-matrices – Minus One-Twelfth Apr 05 '19 at 12:15
  • This is a proof that uses much intuition, the following long formula is copied directly from wikipedia, ($n \times n$ matrices, https://en.wikipedia.org/wiki/Determinant#n_%C3%97_n_matrices)

    $det(A)=\sum_{\sigma \in S_n} sgn(\sigma)\prod_{i=1}^{n}a_{i,\sigma i}$. Note that if $1\le i \le k$ and that $k\le \sigma _i$, then $a{i,\sigma _i}$ must be 0. Hence we can intuitively get the formula.

     –  Apr 05 '19 at 12:15

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